課程名稱︰工程數學上
課程性質︰機械系大二上必修
課程教師︰施文彬
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2006.01.09
考試時限（分鐘）：110 min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Rule: No calculators are allows. You are allowed to bring an A4 size
information sheet. Please provide the details of your calculation.
Good luck!
1. Determine which of the following subsets are subspaces of the indicated
vector spaces, and determine the dimension of each subspace. Explain your
answer, giving proofs or counterexamples.
(a)(5%) The set of all vectors in R^2 with first component equal to 2.
_
(b)(5%) The set of all vectors x =(x1,x2,x3) in R^3 for which x1+x2+x3=0.
2 2 2
(c)(5%) The set of all vectors in R^3 satisfying x1 +x2 +x3 =0.
1
(d)(5%) The set of all functions f(x) in C[0,1] such that ∫f(x)dx =0,
0
where C[0,1] denotes the space of all real valued continuous
functions defined on the closed interval [0,1].
2. Given the system 8x1- x2- x3 = 4
x1+2x2- x3 = 0
2x1- x2+4x3 = 3
_ _ t
(a)(10%) Write this system in the form of Ax =B, where x =(x1,x2,x3) .
Find the AR of A and produce a matrix Ω such that ΩA =AR.
Also, find the dimension of the column space of A.
_
(b)(5%) Find the general solution of Ax =B in vector form.
(c)(5%) Calculate the determinant of A^(-1).
┌ I A ┐
(d)(5%）Let D be the 6x6 matrix given as four 3x3 blocks D =│ │.
└ 0 I ┘
I is the 3x3 identity matrix. 0 is the 3x3 zero matrix.
Find D^(-1).
3.(15%) ┌ b a a a a ┐
│ b b a a a │ -1
Given A =│ b b b a a │. Evaluate det(A ) by first using row
│ b b b b a │
└ b b b b b ┘
reduction to convert A to upper triangular form.
4. ┌ 1 0 0 ┐
Let A =│ 0 1 -1 │
└ 0 -1 1 ┘
(a)(12%) Find all eigenvalues and corresponding engenvectors.
(b)(8%) Find a diagonal matrix D and an orthogonal matrix
-1
such that A =PDP .
10 t
(c)(5%) Find A X where X =(1,1,1)
5. ┌ 1 1 1 ┐ ╭1╮ ╭ 1╮ ╭ 1╮
Given that │ a b c │ has eigenvectors │1│,│ 0│,│-1│.
└ d e f ┘ ╰1╯ ╰-1╯ ╰ 0╯
(a)(10%) Determine the real numbers a,b,c,d,e,f.
(b)(5%) What are the corresponding eigenvalues?
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